1,598 research outputs found
A Matricial Algorithm for Polynomial Refinement
In order to have a multiresolution analysis, the scaling function must be
refinable. That is, it must be the linear combination of 2-dilation,
-translates of itself. Refinable functions used in connection with
wavelets are typically compactly supported. In 2002, David Larson posed the
question in his REU site, "Are all polynomials (of a single variable) finitely
refinable?" That summer the author proved that the answer indeed was true using
basic linear algebra. The result was presented in a number of talks but had not
been typed up until now. The purpose of this short note is to record that
particular proof
Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis
Recently, compressed sensing techniques in combination with both wavelet and
directional representation systems have been very effectively applied to the
problem of image inpainting. However, a mathematical analysis of these
techniques which reveals the underlying geometrical content is completely
missing. In this paper, we provide the first comprehensive analysis in the
continuum domain utilizing the novel concept of clustered sparsity, which
besides leading to asymptotic error bounds also makes the superior behavior of
directional representation systems over wavelets precise. First, we propose an
abstract model for problems of data recovery and derive error bounds for two
different recovery schemes, namely l_1 minimization and thresholding. Second,
we set up a particular microlocal model for an image governed by edges inspired
by seismic data as well as a particular mask to model the missing data, namely
a linear singularity masked by a horizontal strip. Applying the abstract
estimate in the case of wavelets and of shearlets we prove that -- provided the
size of the missing part is asymptotically to the size of the analyzing
functions -- asymptotically precise inpainting can be obtained for this model.
Finally, we show that shearlets can fill strictly larger gaps than wavelets in
this model.Comment: 49 pages, 9 Figure
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